#5227449

order	512
nilpotency class	2
isoclinism	isoclinic class of #5211343
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1817 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(2,4)(5,6)(7,11)(8,9)(10,12)(13,14,18,16)(15,19,21,22)(17,24,25,20)(23,28,27,26)(31,37)(33,41)(35,38)(36,40);(6,9)(7,12)(15,21)(19,22)(23,27)(26,28)(29,30,34,32)(31,38,37,35)(33,40,41,36)(39,42,43,44);(5,6,8,9)(7,10,12,11)(13,15,18,21)(14,19,16,22)(17,23,25,27)(20,26,24,28)(29,31,34,37)(30,35,32,38)(33,39,41,43)(36,42,40,44);(1,2)(3,4)(5,7)(6,10)(8,12)(9,11)(13,16,18,14)(15,22,21,19)(17,24,25,20)(23,28,27,26)(29,32,34,30)(31,35,37,38)(33,40,41,36)(39,42,43,44);(5,6)(7,10)(8,9)(11,12)(13,17,18,25)(14,20,16,24)(15,23,21,27)(19,26,22,28)(29,33)(30,36)(31,39)(32,40)(34,41)(35,42)(37,43)(38,44)

Polycyclic presentation:

2^9|a^2=gh,b^2=fg,c^2=h,d^2=fh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5134645);